Position matters in every round, except maybe at the very end. This is especially true in bigger leagues. For example, I really like Gates and Finley, but in 20 man, with pick 19, I could either drastically reach on one of them with my second pick (which is not smart) or wait, but they aren't going to come back to me by pick #59. Thus, I can't land them. This is true, albeit on a smaller scale, for more realistically-sized leagues. So might only have Gates/Finley on a couple teams, and since TEs are a dime a dozen later on, I might have 4-5 different TEs across 6 leagues. And this is the same for other positions, due to the sheer volume of different RBs and WRs available.
What everyone else decides to do also matters. For example, I rate Kolb, Stafford, and Freeman about the same. Let's say, for the sake of argument, In three separate drafts in leagues with the same number of teams and same settings, I have the same exact starting pick in each as well. Kolb, Stafford and Freeman should go -roughly- around the same time. Let's say my strategy is to land one of these three guys, because my logic is that QB quality falls off drastically afterwards. I look at the teams picking in between my picks and see that two need QBs, so I go ahead and fill another position of need and wait for my next pick. Two of those three guys go, and then in my next pick I grab the last one. Thus, I could end up with three different QBs in three different leagues following the same strategy. And again, this goes for any position.
Lastly, when you fill certain positions absolutely matters in determining your team composition. If you look at a team with Aaron Rodgers and a team with Matt Ryan, you should see two very different teams. Not saying there won't be any overlap, but there shouldn't be much. Taking a QB in round 1 means you are probably getting WRs or RBs in rounds 2 and 3, which are a different pool of players than the WR/RB pool in round 1. And again, this goes across every round, except for the very end.
I think it's a bad idea to go into every draft thinking you should be making the same team over and over again. You need to adapt to the different situations.
EDIT: Also, I'm not saying there shouldn't be any overlap. Especially towards the end of the draft. I have Larry Fitzgerald on a bunch of my teams cause I really like him this year. I also have Tolbert a lot. I have deeper guys on multiple teams as well. But I have pretty different teams otherwise. It's all about grabbing players in round where you think they will exceed their draft value. And since each draft is so different, following that rule, your teams may not end up the same, and that's perfectly fine.
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